High-temperature expansions for the free energy, the susceptibility, and the second correlation moment of the classical
N-vector model [also known as the
O(N) symmetric classical spin-Heisenberg model or as the lattice
O(N) nonlinear $\ensuremath{\sigma}$ model] on the simple-cubic and the body-centered-cubic lattices are extended to order ${\ensuremath{\beta}}^{21}$ for arbitrary
N. The series for the second field derivative of the susceptibility is extended to order ${\ensuremath{\beta}}^{17}$. We report here on the analysis of the computed series for the susceptibility and the (second moment) correlation length which yields updated estimates of the critical parameters for various values of the spin dimensionality
N, including
N=0 (the self-avoiding walk model),
N=1 (the Ising spin-1/2 model),
N=2 (the
XY model), and
N=3 (the classical Heisenberg model). For all values of
N we confirm a good agreement with the present renormalization-group estimates. A study of the series for the other observables will appear in a forthcoming paper.